According to Serre in "Zeta and L-functions" (MR0194396), these L-functions have been defined by Artin himself. Moreover, it seems that in the formula $$ L(s,\rho) = \prod_{\substack{\text{closed points} \\ x \in X}}\mathrm{det}\left(I - \frac{\rho(\mathrm{Frob}_x)}{N(x)^s}\right)^{-1} $$ one should define $\rho(\mathrm{Frob}_x)$ as the average of the $\rho(g)$ for $g \in G$ mapping to $\mathrm{Frob}_x$.

Serre further says that $L(s,\rho)$ can be continued to a meromorphic function in the half-plane $\mathrm{Re}(s)>n-\frac12$, and that the singularities « can be determined, or rather reduced to the classical case $n=1$ », but I don't know whether this answers your Question 2.

Artin's paper is Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren. Abh. Math. Sem. Univ. Hamburg 8 (1931), no. 1, 292--306 (MR3069563).

According to Serre in "Zeta and L-functions" (MR0194396), these L-functions have been defined by Artin himself. Moreover, it seems that in the formula $$ L(s,\rho) = \prod_{\substack{\text{closed points} \\ x \in X}}\mathrm{det}\left(I - \frac{\rho(\mathrm{Frob}_x)}{N(x)^s}\right)^{-1} $$ one should define $\rho(\mathrm{Frob}_x)$ as the average of the $\rho(g)$ for $g \in G$ mapping to $\mathrm{Frob}_x$.

Serre further says that $L(s,\rho)$ can be continued to a meromorphic function in the half-plane $\mathrm{Re}(s)>n-\frac12$, and that the singularities « can be determined, or rather reduced to the classical case $n=1$ » (there is an interesting lemma in the case $Y \to Y'$ is a fibration of curves, enabling to do induction on the dimension of $Y$). I don't know whether this answers your Question 2.

Artin's paper is Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren. Abh. Math. Sem. Univ. Hamburg 8 (1931), no. 1, 292--306 (MR3069563).